C7X's Structural Array Notations
Use the operator \(:=\) to represent assignment/definition. \(S_1-S_2\) is the relative complement of sets \(S_1\) and \(S_2\). Epsilon Array Notation Definition Create a set \(\text{Ar}\): \(0 \in \text{Ar} \\ A_1\in\text{Ar} \implies A_1 \in \text{Ar} \\ A_1,A_2\in\text{Ar} \implies A_1,A_2 \in \text{Ar}\) Create a set \(\text{Br}\): \(A_1\in(\text{Ar}-0) \implies A_1 \in \text{Br} \\ A_1,A_2\in\text{Br} \implies A_1,A_2 \in \text{Br} \) Define a relation \(=\) over \(\text{Ar}\): \(0=0 \\ A_1=A_2\iff A_1=A_2 \\ A_1,A_2=A_3,A_4\iff A_1=A_3\land A_2=A_4\) Define \(*(A_1,A_2,n)\) where \(A_1,A_2\in\text{Ar}\) and \(n\in\mathbb{N}\): \(*(A_1,A_2,0) := A_1 \\ *(A_1,A_2,n+1) := *(A_1,A_2,n),A_2\) Define a partial function \(\text{Reduce}(A,n)\) where \(A\in\text{Br}\) and \(n\in\mathbb{N}\): \(A=[ 0] \implies \text{Reduce}(A,n) := *(0,0,n) \\ A=[ [A_1,0]]:A_1\in\text{Ar} \implies \text{Reduce}(A,n) := *(A_1,A_1,n) \\ A=A_1:A_1\in\text{Br} \implies \text{Reduce}(A,n) := \text{Reduce}(A_1,n) \\ A=[A_1,A_2]:A_1\in\text{Br}\land A_2\in(\text{Ar}-0) \implies \text{Reduce}(A,n) := *(A_1,\text{Reduce}(A_2,n),n) \\ A=[A_1,A_2,A_3]:A_1,A_2,A_3\in\text{Br} \implies \text{Reduce}(A,n) := [A_1,A_2,\text{Reduce}(A_3,n)]?\) Define \(\varepsilon(A,n)\) where \(A\in\text{Ar}\), and \(n\in\mathbb{N}\): \(A=0 \implies \varepsilon(A,n) := n+1 \\ A=[A_1,0]:A_1\in\text{Ar} \implies \varepsilon(A,n) := \underbrace{\varepsilon(A_1,\cdots\varepsilon(A_1,\varepsilon(}_{n}A_1,n))\cdots) \\ A\in\text{Br}\implies \varepsilon(A,n) := \varepsilon(\text{Reduce}(A,n),n)\) Explanation Explanation is incorrect/incomplete A "valid array" is \(0\), \(A\), or \(A,B\) where \(A\) and \(B\) are valid arrays. A "singular array" is \(A,B\) where \(A\) and \(B\) are singular arrays, or \(A\) where \(A\) is a valid array other than \(0\). The singular arrays are a subset of the valid arrays. For the purposes of this explanation, use \(1\) as a shorthand for \(0\). Define an array notation \(\{A\}n\) for a singular array \(A\) and a natural number \(n\): \(\{[ 0]\}n=n\) \(\{A,[0]\}n=\underbrace{[ [ \cdots [ [ }_{n}A,A],A] \cdots,A],A]\) where \(A\) is valid \(\{A\}n=\{A,n\}\) where \(A\) is singular \(\{A,B\}n=[A,\{B\}n]\) where \(A\) and \(B\) are singular Now define \(\varepsilon(A,n)\) for valid array \(A\) and natural number \(N\): \(\varepsilon(0,n)=n+1\) \(\varepsilon(A,1,n)=\underbrace{\varepsilon(A,\varepsilon(A,\cdots\varepsilon(}_{n}A,n)\cdots))\) \(\varepsilon(A,n)=\varepsilon(P(\{A\}n),n)\) if \(A\) is singular Examples \(\varepsilon([ 0],2) \\ =\varepsilon(\text{Reduce}([ 0],2),2)\text{ (}\varepsilon\text{ Rule 3)} \\ =\varepsilon(*(0,0,2),2)\text{ (Reduce Rule 1)} \\ =\varepsilon([*(0,0,1),0],2)\text{ (* Rule 2)} \\ =\varepsilon([ [*(0,0,0),0],0],2)\text{ (* Rule 2)} \\ =\varepsilon([ [ 0,0],0],2)\text{ (* Rule 1)} \\ =\varepsilon([ 0,0],\varepsilon([ 0,0],2))\text{ (}\varepsilon\text{ Rule 2)}\) \(\) Analysis \(\varepsilon(0,n)\approx f(0,n) \\ \varepsilon(0,n)\approx f(1,n) \\ \varepsilon([ 0,0],n)\approx f(2,n) \\ \varepsilon([ [ 0,0],0],n)\approx f(3,n) \\ \varepsilon([ 0],n)\approx f(\omega,n) \\ \varepsilon([ [ 0],0],n)\approx f(\omega+1,n) \\ \varepsilon([ [ [ 0],0],0],n)\approx f(\omega+2,n) \\ \varepsilon([ [ [ [ 0],0],0],0],n)\approx f(\omega+3,n) \\ \varepsilon([ [ 0],[ 0]],n)\approx f(\omega2,n) \\ \varepsilon([ [ [ 0],[ 0]],0],n)\approx f(\omega2+1,n) \\ \varepsilon([ [ [ [ 0],[ 0]],0],0],n)\approx f(\omega2+2,n) \\ \varepsilon([ [ [ 0],[ 0]],[ 0]],n)\approx f(\omega3,n) \\ \varepsilon([ [ [ [ 0],[ 0]],[ 0]],0],n)\approx f(\omega4,n) \\ \varepsilon([ [ 0,0]],n)\approx f(\omega^2,n) \\ \varepsilon([ [ [ 0,0]],[ 0]],n)\approx f(\omega^2+\omega,n) \\ \varepsilon([ [ [ 0,0],0]],n)\approx f(\omega^3,n) \\ \varepsilon([ [ 0]],n)\approx f(\omega^\omega,n) \\ \varepsilon([ [ [ 0]],0],n)\approx f(\omega^\omega+1,n) \\ \varepsilon([ [ [ 0]],[ 0]],n)\approx f(\omega^\omega+\omega,n) \\ \varepsilon([ [ [ 0]],[ [ 0]]],n)\approx f(\omega^\omega2,n) \\ \varepsilon([ [ [ 0],[ 0]]],n)\approx f(\omega^{\omega2},n) \\ \varepsilon([ [ [ 0]]],n)\approx f(\omega^{\omega^\omega},n) \\ \varepsilon([ [ [ [ 0]]]],n)\approx f(\omega^{\omega^{\omega^\omega}},n)\) Zeta Array Notation Definition Create a set \(\text{Ar}\): \(0 \in \text{Ar} \\ A\in\text{Ar} \implies A \in \text{Ar} \\ A_1,A_2\in\text{Ar} \implies A_1,A_2 \in \text{Ar} \\ A\in\text{Ar} \implies \{A\}\in\text{Ar}\) Create a set \(\text{Br}\): \(A\in(\text{Ar}-0) \implies A \in \text{Br} \\ A_1,A_2\in\text{Br} \implies A_1,A_2 \in \text{Br} \\ A\in\text{Ar} \implies \{A\}\in\text{Br} \) Define a relation \(=\) over \(\text{Ar}\): \(0=0 \\ A_1=A_2\iff A_1=A_2 \\ A_1,A_2=A_3,A_4\iff A_1=A_3\land A_2=A_4 \\ \{A_1\}=\{A_2\}\iff A_1=A_2\) Define \(*(A_1,A_2,n)\) where \(A_1,A_2\in\text{Ar}\) and \(n\in\mathbb{N}\): \(*(A_1,A_2,0) := A_1 \\ *(A_1,A_2,n+1) := *(A_1,A_2,n),A_2\) Define a partial function \(\text{Reduce}(A,n)\) where \(A\in\text{Br}\) and \(n\in\mathbb{N}\): \(A=[ 0] \implies \text{Reduce}(A,n) := *(0,0,n) \\ A=\{0\} \implies \text{Reduce}(A,n) := \underbrace {[ [ [}_{n}0]\cdots]] \\ A=[ [A_1,0]]:A_1\in\text{Ar} \implies \text{Reduce}(A,n) := *(A_1,A_1,n) \\ A=A_1:A_1\in\text{Br} \implies \text{Reduce}(A,n) := \text{Reduce}(A_1,n) \\ A=\{[A_1,0]\}:A_1\in\text{Ar} \implies \text{Reduce}(A,n) := \underbrace{[ [\cdots[ [}_{n}[\{A_1\},0]]]\cdots]] \\ A=\{A_1\}:A_1\in\text{Br} \implies \text{Reduce}(A,n) := \{\text{Reduce}(A_1,n)\} \\ A=A_1,\{A_2\}:A_1\in\text{Br}\land A_2\in\text{Ar} \implies \text{Reduce}(A,n) := A_1,\text{Reduce}(\{A_2\},n)? \\ A=[A_1,A_2]:A_1\in\text{Br}\land A_2\in(\text{Ar}-0) \implies \text{Reduce}(A,n) := *(A_1,\text{Reduce}(A_2,n),n) \\ A=[A_1,A_2,A_3]:A_1,A_2,A_3\in\text{Br} \implies \text{Reduce}(A,n) := [A_1,A_2,\text{Reduce}(A_3,n)]?\) Define \(\zeta(A,n)\) where \(A\in\text{Br}\) and \(n\in\mathbb{N}\): \(A=0 \implies \zeta(A,n) := n+1 \\ A=[A_1,0]:A_1\in\text{Ar} \implies \zeta(A,n) := \underbrace{\zeta(A_1,\cdots\zeta(A_1,\zeta(}_{n}A_1,n))\cdots) \\ A\in\text{Br} \implies \zeta(A,n) := \zeta(\text{Reduce}(A,n),n)\) Analysis \(\zeta(0,n)\approx f(0,n) \\ \zeta(0,n)\approx f(1,n) \\ \zeta([ 0,0],n)\approx f(2,n) \\ \zeta([ [ 0,0],0],n)\approx f(3,n) \\ \zeta([ 0],n)\approx f(\omega,n) \\ \zeta([ [ 0],0],n)\approx f(\omega+1,n) \\ \zeta([ [ [ 0],0],0],n)\approx f(\omega+2,n) \\ \zeta([ [ [ [ 0],0],0],0],n)\approx f(\omega+3,n) \\ \zeta([ [ [ 0,0],0]],n)\approx f(\omega^3,n) \\ \zeta([ [ 0]],n)\approx f(\omega^\omega,n) \\ \zeta([ [ [ 0]],0],n)\approx f(\omega^\omega+1,n) \\ \zeta([ [ [ 0]],[ 0]],n)\approx f(\omega^\omega+\omega,n) \\ \zeta([ [ [ 0]],[ [ 0]]],n)\approx f(\omega^\omega2,n) \\ \zeta([ [ [ 0],[ 0]]],n)\approx f(\omega^{\omega2},n) \\ \zeta([ [ [ 0]]],n)\approx f(\omega^{\omega^\omega},n) \\ \zeta([ [ [ [ 0]]]],n)\approx f(\omega^{\omega^{\omega^\omega}},n) \\ \zeta(\{0\},n)\approx f(\varepsilon_0,n) \\ \zeta([\{0\},0],n)\approx f(\varepsilon_0+1,n) \\ \zeta([\{0\},[ 0]],n)\approx f(\varepsilon_0+\omega,n) \\ \zeta(\{0\},\{0\},n)\approx f(\varepsilon_02,n) \\ \zeta([ \{0\},\{0\},\{0\}],n)\approx f(\varepsilon_03,n) \\ \zeta([ [\{0\},0],n)\approx f(\omega^{\varepsilon_0+1},n)=f(\varepsilon_0\omega,n) \\ \zeta(\{0\},n)\approx f(\varepsilon_1,n) \\ \zeta(\{[ 0]\},n)\approx f(\varepsilon_\omega,n) \\ \zeta(\{\{0\}\},n)\approx f(\varepsilon_{\varepsilon_0},n) \\ \zeta(\{\{\{0\}\}\},n)\approx f(\varepsilon_{\varepsilon_{\varepsilon_0}},n) \) Gamma Array Notation Ill-defined Definition Create a set \(\text{Ar}\): \(0 \in \text{Ar} \\ A_1,A_2\in\text{Ar} \implies A_1,A_2 \in \text{Ar} \\ A_1,A_2\in\text{Ar} \implies \{A_1,A_2\}\in\text{Ar}\) Create a set \(\text{Br}\): \(A_1,A_2\in\text{Br} \implies A_1,A_2 \in \text{Br} \\ A_1\in\text{Ar}\land A_2\in(\text{Ar}-0) \implies \{A_1,A_2\} \in \text{Br}\) Define a relation \(=\) over \(\text{Ar}\): \(0=0 \\ A_1,A_2=A_3,A_4\iff A_1=A_3\land A_2=A_4 \\ \{A_1,A_2\}=\{A_3,A_4\}\iff A_1=A_3\land A_2=A_4\) Define \(*(A,n)\) where \(A\in\text{Ar}\) and \(n\in\mathbb{N}\): \(*(A,1) := A \\ *(A,n+1) := *(A,n),A\) Define \(P(A)\) where \(A\in\text{Ar}\): \(A=[A_1,[A_2,0]]:A_1,A_2\in\text{Ar} \implies P(A) := P(A_1,[0,A_2]) \\ A=[A_1,0]:A_1\in\text{Ar} \implies P(A) := A\) Define a partial function \(\text{Reduce}(A,n)\) where \(A\in\text{Br}\) and \(n\in\mathbb{N}\): \(A=\{0,\{0,0\}\} \implies \text{Reduce}(A,n) := *(\{0,0\},n) \\ A=\{A_1,\{0,0\},0\} \implies \text{Reduce}(A,n) := \underbrace{\{n,\{n,\cdots \{n,\{}_{n}n,0\}\}\cdots\}\} \\ A=\{0,A_1,\{0,0\}\}:A_1\in\text{Ar} \implies \text{Reduce}(A,n) := *(A_1,n) \\ A=\{A_1,\{0,0\},A_2,\{0,0\}\}:A_1,A_2\in\text{Ar} \implies \text{Reduce}(A,n) := \underbrace{\{A_1,\{A_1,\cdots\{A_1,\{A_1,}_{n}\{A_1,A_2\},\{0,0\}\}\}\cdots\}\} \\ A=\{A_2,A_1\}:A_1\in\text{Br}\land A_2\in\text{Ar} \implies \text{Reduce}(A,n) := \{A_2,\text{Reduce}(A_1,n)\} \\ A=A_1,A_2:A_1,A_2\in\text{Br} \implies \text{Reduce}(A,n) := A_1,\text{Reduce}(A_2,n)\) Define \(\Gamma(A,n)\) where \(A\in\text{Br}\) and \(n\in\mathbb{N}\): \(A=0 \implies \Gamma(A,n) := n+1 \\ A=[A_1,0]:A_1\in\text{Ar} \implies \Gamma(A,n) := \underbrace{\Gamma(A_1,\cdots\Gamma(A_1,\Gamma(}_{n}A_1,n))\cdots) \\ A\in\text{Br} \implies \Gamma(A,n) := \Gamma(P(\text{Reduce}(A,n)),n)\) Analysis SoonCategory:ARRAY NOTATIONS Category:C7X's Stuff Category:POTENTIALLY SOMEHOW-DEFINED